Linear dependence and independence IV Determine the vector collection a1=(1, 1, 1), 02=(0, 2, 5),a3=(1, 3, 6) are linearly independent in R3? tion a1(1,1,1)7+c2(0,2,5)7+c3(1,3,6)7=(0,0.0)7, 0 C1+2c2+3c3=0 C1+5c2+6c3 The coefficient matrix of this system 101 101 A=(a1,a2,a3) 156 055Linear dependence and independence IV Example Determine the vector collection α1 = (1, 1, 1)T , α2 = (0, 2, 5)T , α3 = (1, 3, 6)T are linearly independent in R 3 ? Solution: If c1(1, 1, 1)T + c2(0, 2, 5)T + c3(1, 3, 6)T = (0, 0, 0)T , then c1 + c3 = 0 c1 + 2c2 + 3c3 = 0 c1 + 5c2 + 6c3 = 0. The coefficient matrix of this system A = (α1, α2, α3) =   1 0 1 1 2 3 1 5 6   r2−r1 r3−r1 −→   1 0 1 0 2 2 0 5 5   r3− 5 2 r2 −→   1 0 1 0 2 2 0 0 0   () May 3, 2006 4 / 40